Björner–Las Vergnas–Sturmfels–White–Ziegler conjecture on bichromatic triangles

An Euclidean pseudoline arrangement is a finite collection of bi-infinite, simple curves in the Euclidean plane such that every pair crosses exactly once; it is simple if no three pseudolines meet at a common point. A bicolored arrangement is an arrangement whose pseudolines are colored blue and red, with at least one pseudoline of each color. A bichromatic triangle is a triangular face supported by pseudolines of both colors.

Björner–Las Vergnas–Sturmfels–White–Ziegler conjecture. Every bicolored arrangement has a bichromatic triangle.

This question was posed in 1993. The paper states that the conjecture is solved when there are at most five red pseudolines, but leaves the general case open.